Magnetic force is caused by the movement of charge. However, it is more fully described as a manifestation of a single electromagnetic field. This field can appear as an elctrostatic field or a magnetic field depending on one's frame of reference. The properties are described by Maxwell's equations and these equations are invariant under a Lorentz transformation and so are consistent with special relativity.

I'm not sure whether a simpler or a more thorough explanation is required.

Magnetic force is caused by the movement of charge. However, it is more fully described as a manifestation of a single electromagnetic field. This field can appear as an elctrostatic field or a magnetic field depending on one's frame of reference. The properties are described by Maxwell's equations and these equations are invariant under a Lorentz transformation and so are consistent with special relativity.

I'm not sure whether a simpler or a more thorough explanation is required.

Yes, there are moving charges in a permanent magnet. Atoms themselves can be visualised as electrons rotating about a positive nucleus and they can have a dipole magnetic field associated with them. If you can align these atoms in a similar orientation then there can result a significant magnetic field. In permanent magnets atoms get aligned in this way.

The structures can be more complicated as groups of material with a statistical imbalance of alignments (producing a magnetic field) may be forced into alignment with other "domains" with a similar orientation by mechanical or outside magnetic forces.

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